How modify sphere mesh in an animation with shape keys
You could simply use a Displace Modifier and apply it as a shapekey.

First add a Subdivision Surface Modifier followed by the Displace Modifier and then a second Subdivision Surface Modifier.
Your modifier stack should look like the image below:

The texture can be anything grayscale, but for this example I used the below:

Once you are happy with the result of the texture you choose, apply the first Subdivision Surface Modifier, and apply the Displace Modifier as a shape key.

The remaining Subdivision Surface Modifier will not be able to be applied, and shouldn't be anyway.